# Parallelogram Law of Vector Addition

**State Parallelogram Law of vector addition**

**If a parallelogram can be drawn so that two vectors can be placed with their tails connected as the two adjacent sides of the parallelogram with an angle θ between them, then the diagonal of the parallelogram represents their resultant vector or the vector sum.**

**Derive the formulas of the resultant vector** using the **Parallelogram Law of vector addition**

Let **A** and **B** be the two vectors and let θ be the angle between them as shown in Fig. above. To calculate the vector sum, we complete the parallelogram.

Here side PQ represents vector A, side PS represents B and the diagonal PR represents the resultant vector R. Here, α angle is the angle the resultant makes with the base vector and the angle denotes the direction of the resultant or the vector sum.

**Sample Numerical to find the sum of 2 force vectors using the Parallelogram Law**

**Force A has a magnitude of 30 N and it makes an angle of 30 degrees with another force B of magnitude 40 N. What is the vector sum or resultant of forces A and B?**

**Solution**:

Let,

Rbe the vector sum of forcesAandB.|R| = √[30

^{2}+ 40^{2}+ 2.30.40 cos 30] =67.66NIf resultant force R makes an angle α with A then

tan α = (B sinθ)/(A + Bcosθ)

here, A = 30, B = 40, θ = 30.tanα = 40 sin 30 / (30 + 40 cos 30) = 20/(30 + 34.64) = .31

α = arctan (.31) =

17.22 degreeAnswer: The resultant force is 67.66 N at 17.22 degrees

from forceA